Novel shadows from the asymmetric thin-shell wormhole

Xiaobao Wang 1,∗ Peng-Cheng Li2,3,† Cheng-Yong Zhang4,‡ and Minyong Guo 2§1 School of Applied Science,

Beijing Information Science and Technology University,Beijing 100192, P. R. China

2Center for High Energy Physics, Peking University,No.5 Yiheyuan Rd, Beijing 100871, P. R. China

3Department of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University, No.5 Yiheyuan Rd,

Beijing 100871, P.R. China4 Department of Physics and Siyuan Laboratory,

Jinan University, Guangzhou 510632, China

For dark compact objects such as black holes or wormholes, the shadow size has long been thoughtto be determined by the unstable photon sphere (region). However, by considering the asymmetricthin-shell wormhole (ATSW) model, we find that the impact parameter of the null geodesics isdiscontinuous through the wormhole in general and hence we identify novel shadows whose sizesare dependent of the photon sphere in the other side of the spacetime. The novel shadows appearin three cases: (A2) The observer’s spacetime contains a photon sphere and the mass parameter issmaller than that of the opposite side; (B1, B2) there’ s no photon sphere no matter which massparameter is bigger. In particular, comparing with the black hole, the wormhole shadow size isalways smaller and their difference is significant in most cases, which provides a potential way toobserve wormholes directly through Event Horizon Telescope with better detection capability in thefuture.

Introduction.— The image of M87* taken by Event Hori-zon Telescope made its debut and revealed the shadowof black holes in April 2019 [1], which has ignited a waveof researches into the shadows of black holes. As is well-known, the shadow size of a black hole is considered to bedetermined by the unstable photon sphere (region) otherthan the event horizon from the earliest work [2, 3]. Asone of the most important predictions of general relativ-ity (GR) besides black holes, wormhole spacetime [4] alsocontains unstable photon sphere which also creates shad-ows which were found to be different from the black holesin the size or the oblateness, see examples in [5–10]. Eventhough it has been shown a traversable wormhole withordinary matter is not allowed in GR, if there is some ex-otic matter in the universe, it becomes possible [11, 12].Furthermore, if the exotic matter distributes into a thinshell, a simple model of thin-shell wormhole can be con-structed by the “cut and paste” technique [13, 14]. Andas far as we know, the shadow of a asymmetric thin-shellwormhole has not been studied before in general.

In this letter, we focus on an asymmetric thin-shellwormhole (ATSW) spacetime, more specially, a staticthin shell connecting two distinct Schwarzschild space-times whose difference is only the mass parameter [14].The Generic spherically symmetric dynamic thin-shelltraversable wormholes and their stabilities were discussed

∗ bao@mail.bnu.edu.cn† lipch2019@pku.edu.cn‡ zhangcy@email.jnu.edu.cn§ Corresponding author:minyongguo@pku.edu.cn

in [15]. Using the backward ray-tracing method, we care-fully analyze the ingoing null particles that emit fromthe observer’ s position in one side and pass through thethroat to the opposite side of the thin shell. Note that thestatic observers on both sides of the spacetime is differ-ent, thus we first find the impact parameter of the samephoton defined in each side is different and related by asimple equation. Correspondingly, the turning point ineach side for the same photon becomes different. Con-sidering a physical process that ingoing photons with noturning point would fall into the thin shell and then turninto outgoing in the other side, we first find some of themmay hit its turning point defined in this spacetime andthen turn back to its birthplace. Thus, we can see thedifference between the traversable wormhole and blackhole from the null geodesic motion. It’s worth notingthat for a black hole, the light that enters the event hori-zon never returns, since the event horizon is a “one-way”membrane.

Based on our new finding, we identify novel shadowsin ATSW spacetime (see [16] for a similar study ). Inparticular, these novel shadows originate from spacetimeasymmetry of both sides connected by the throat andtheir sizes do not depend on the photon sphere, contraryto what we typically think about. Specifically, let’s as-sume we stay in the spacetimeM1 and the radius of thethin shell is R. See Fig. 1, the shadow of the wormholeseen by the static observer is the same with that of thecorresponding black hole when M1 contains the photon

sphere R ≤ rph1 and M1 > M2, that is, there is no novelshadow and the throat behaves like an “event horizon”. However, we find novel shadows in other cases. When

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FIG. 1. The shadow of thin-shell wormhole for various pa-rameters. There is novel shadow for some situations.

R ≤ rph1 and M2 > M1, the shadow size of the wormhole

becomes rsh1 = 3√

3M2

√R−2M2

R−2M1which is determined by

the photon sphere of the opposite side and always smallerthan that in Schwarzschild spacetime since the photonswith the impact parameter b1 < 3

√3M1 would fall into

the spacetimeM2 and turn back toM1 which never oc-curs for black hole. In particular, the size of the shadowcan range from zero to 3

√3M1, so that for some param-

eters the difference can be large enough to be detectedto distinguish the wormhole from the black hole. When

there is no photon sphere in M1, that is, R > rph1 , sur-prisingly, there also exists a novel shadow. For M1 > M2

since R is the turning point for certain null geodesics withthe corresponding impact parameter bR1 and the photonswith b1 < bR1 will go through the thin shell and never turnback, the shadow of the wormhole depends on the thinshell and its radius is rsh1 = R√

f1(R). While for M2 > M1

and max2M2, 3M1 < R < 3M2, the side M2 containsthe photon sphere and some null geodesics with b1 < bR1could turn back after arriving at the spacetimeM2, thus

the critical impact parameter is 3√

3M2

√R−2M2

R−2M1and

the novel shadow size is rsh1 = 3√

3M2

√R−2M2

R−2M1which

is smaller than bR1 . Interestingly, this expression is the

same with that for M2 > M1 and R ≤ rph1 , see Fig. 1.

Background of Schwarzschild black hole shadow and thethin shell wormhole.—Let us start from some importantphysical quantities and useful symbols in terms of theshadow in the background of Schwarzschild black holewhose metric takes in this form

ds2 = −f(r)dr2 + f−1(r)dr2 + r2dΩ2, (1)

where f(r) = 1− 2M/r. Considering a null particle withmomentum vector pa, the radial motion of the geodesicis described by the equation

pr = ±E√

1− b2

r2f(r), (2)

FIG. 2. The left picture shows behaviors of the func-tion r3(pr)2 with three different kinds of impact parameters.There’s no turning point for b < bc, single turning point forb = bc and two turning points for b > bc. The right onepresents the shadow of Schwarzschild black hole.

where b = L/E is the impact parameter and ± standsfor outgoing and ingoing directions, respectively. Theequation pr = 0 gives b2 = r2/f(r) has two real roots

for the positive branch of r when b ≥ bc = 3√

3M . Forb = bc, both roots are equal to rph = 3M is knownas the radius of photon sphere. For b > bc, the rootsare addressed as the turning points with the larger rootrl > rph and the smaller one 2M = rh < rs < rph, whererh is the radius of the event horizon. Also, we introducea new parameter bl,s ≡ rl,s/

√f(rl,s) that will be later

used. Moreover, considering a distant static observer at(to, ro → ∞, π/2, 0), the angular radius of the shadowleft by the photon sphere satisfies sin θ = bc/ro and theradius of the shadow is defined by rsh ≡ ro sin θ = bc, asseen in Fig. 2.

FIG. 3. The diagram of the thin-shell wormhole spacetime.In this figure, we give an example that in M1 an ingoingphoton with certain impact parameter pass through the thinshell and stops at its turning point in M2 and turns back toM1.

Next, let’s introduce some necessary backgroundknowledge of thin-shell wormhole using cut-and-pastemethod, that is, two distinct spacetimes M1,2 with dif-ferent parameters are glued by a thin shell which formsa new manifold M = M1 ∪M2, as seen in Fig. 3. Wesuppose that a spherical thin shell is moving in a spheri-cally symmetric spacetime and the metrics on both sides

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take in this form

ds2i = −fi(ri)dt2i + f−1i (ri)dr2i + r2i dΩ2, (3)

where i = 1, 2, and by focusing on the Schwarzschildcase we have fi(r) = 1 − 2Mi

ri, where Mi are the mass

parameters.The local tetrads in the neighbourhood of the thin shell

of each spacetime M1,2

eati ≡ f− 1

2i (R)

(∂

∂ti

)a, (4)

eari ≡√fi(R)

(∂

∂ri

)a, (5)

are related by the following Lorentz transformation [17](eat2ear2

)= Λ(Θ1 −Θ2)

(eat1ear1

). (6)

where we have defined

Λ(Θ) =

(cosh Θ sinh Θsinh Θ cosh Θ

), (7)

with

Θi = sinh−1εiR√fi(R)

. (8)

Here R is the radius of the thin shell and R denotes thevelocity of the moving thin shell.

Conserved quantities and some conventions—Now, weconsider an ingoing photon from the spacetimeM1 pass-ing through the thin shell. For simplicity, we assume theinteraction between the photon and the thin shell is onlygoverned by gravity which implies the 4-momentum pa

is invariant during the process passing through the thinshell. The similar treatment has been applied in [18] Inaddition, the metric of the spacetime M is continuousby definition which means gM1

ab (R) = gM2

ab (R). Hence,the conserved quantities pti = −Ei and pφi

= Li along ageodesic imply

− E2√f2(R)

= −cosh (Θ1 −Θ2)√f1(R)

E1 +sinh (Θ1 −Θ2)√

f1(R)pr1R ,

L1 = L2, (9)

at the position of the thin shell, and here we have usedpr1R to denote the value of pr1 on the thin shell. UsingEq. (2), Eq. (9) can be simplified as

b1b2

=

√f2(R)

f1(R), (10)

where for simplicity we have set R = 0 without losing thephysics nature, that is, we assume the thin-shell worm-hole is static which can be proven that if we tolerate the

violation of null energy condition, we could always findthe thin-shell wormhole to be static. From this equation,we find b1 = b2 when f1(R) = f2(R), that is M1 = M2,which tells us that the spacetime M is symmetric aboutthe thin shell. However, if b1 6= b2, the mass parametersofM1 andM2 are no longer equal and the spacetimeMbecomes asymmetric about the thin shell which we focuson in this letter.

Before we move to discuss the shadow of the asymmet-ric thin-shell wormhole, in order to calculate convenientlywe would like to establish some conventions. Let’s sup-pose M1 = 1 and M2 = k > 0 for the mass parameter.Thus we have R > max2, 2k for thin-shell wormhole.Based on the previous review of the background of theshadow, the radius of the photon sphere ofM1 andM2 is

rph1 = 3 and rph2 = 3k, respectively. Correspondingly, the

critical impact parameter reads bc1 = 3√

3 and bc2 = 3√

3kfor each side. Furthermore, as mentioned before we usebR1,2 = R√

f1,2(R)to denote the specific impact parameter

in which case one finds pR1,2 = 0. Using these conventions,Eq. (10) can be rewritten as

b1b2

=

√R− 2k

R− 2=

√1− 2(k − 1)

R− 2. (11)

The size of the thin-shell wormhole—Now, we directlyface the calculation of the thin-shell wormhole shadow.Unlike horizons of black holes, the throat of thin-shellwormhole is not fixed, instead, the thin-shell throatonly needs to be larger than the horizon of the blackhole in the corresponding spacetime. Thus, whether thephoton sphere exists in M1 is unknown, that is, bothmax2k, 2 < R ≤ 3 and R > 3 are possible, but if thelatter is true, wormholes are even more different fromblack holes in terms of null geodesic’s motion. There-fore, we would like to discuss the former first.(A1). Since the relationship of size between M1 and

M2 is not clear, but it matters a lot. We might as wellconsider 0 < k < 1 < R/2 ≤ 3/2 now. In addition,for the geodesics with b1 ≥ bc1, the known results inSchwarzschild spacetime containing a black hole can beapplied to the thin-shell wormhole. Next, we carefullydiscuss the ingoing photons with b1 < bc1 = 3

√3, we find

b2 <

√R− 2√R− 2k

3√

3 ≡ Bc2. (12)

Furthermore, since R > 2 > 2k, there are always nullgeodesics with bR2 such that pr2R = 0. From simpleanalysis, we find b2 < Bc2 ≤ bR2 is always true for2 < R ≤ 3. Hence, when R > 3k, R is the largerreal root making pR2 = 0 which implies all the outgoingnull geodesics with b2 < bR2 starting from the thin shellwill go to the infinity in M2. In the range 0 < k < 1,the region that doesn’t satisfy the condition R > 3k is2/3 < k < 1 and 2 < R < 3k, where R is the smaller realroot. In this situation, one expects the outgoing photons

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bc2 < b2 < bR2 will stop at the turning point in the space-time M2 and bounce back to the M1, thus the radiusof a novel shadow of the thin-shell wormhole would be

rsh1 = 3√

3k√

R−2kR−2 ≡ B

c1. However, after some algebraic

calculus, we find Bc2 < bc2 always holds for 2/3 < k < 1and 2 < R < 3k, which tells us there is no turning pointfor null geodesics with b2 < Bc2. From the above, weconclude that when M1 > M2 and the side M1 containsa photon sphere, the shadow of the thin-shell wormholeobserved by the static observer is the same with the cor-responding black hole.

(A2). Next, we turn to the case b1/b2 < 1, that is,

1 < k < R/2, we have R ≤ 3 < 3k = rph2 which impliesthe photon sphere always exists in the spacetime M2.Considering the ingoing null geodesics with b1 < 3

√3 in

the spacetimeM1, if we wish some geodesics would turnback passing through the throat, a necessary condition

b2 =

√R− 2

R− 2kb1 > bc2, (13)

must be hold. Thus, we need

Bc1 =

√R− 2k

R− 23√

3k < b1 < bc1, (14)

holds. Thus, the key is to check if this condition can besatisfied given 1 < k < R

2 and 2 < R < 3. Note that

Bc1 is a decreasing function of k in the range k ∈(1, R2

),

we have 0 < Bc1 < bc1, that is to say, the desired con-dition is always satisfied. Therefore, we find that for1 < k < R

2 ≤ 3/2, the angular radius of the shadow of thethin shell is always smaller than that of the correspond-ing black hole, see Fig. 4. In particular, for R ∼ 2k, theradius of the shadow can be arbitrarily small. In terms ofthe novel shadow induced by the asymmetric mechanismof the thin-shell wormhole, the observational size of theshadow in the spacetime M1 can be very different be-tween wormholes and black holes which may be detectedby the EHT directly, if the resolution is raised to a goodenough level.

FIG. 4. The novel shadow for (A2) case, the spacetime M1

contains the photon sphere and the mass parameter of M1

is smaller than that of M2, namely, 1 < k < R/2 ≤ 3/2. Itcan be seen that the shadow of wormhole is smaller than thatof black hole compared to the right graph in Fig. 2. In fact,from our analysis, the novel shadow of wormhole is alwayssmaller and could be arbitrarily small.

(B1). Now, we move to the case R > 3, and alsofocus on 0 < k < 1 at first. Since R > 3, R must be aouter turning point for a class of null geodesics with theimpact parameter bR1 = R√

f1(R). And the incident null

geodesics will bounce back to the infinity when b1 ≥ bR1in the spacetime M1.

For b1 < bR1 , note bR1 is an increasing function of R,we have bR1 > bc1. When the null geodesics pass throughthe wormhole throat and arrive at the spacetime M2

we have b2 <√

R−2R−2k b

R1 = bR2 . Also, we find bR2 is an

increasing function of R for 0 < k < 1, and obtain bR2 >

bc2. Combining with the condition R > 3k = rph2 weconclude the outgoing geodesics with b2 < bR2 will goto infinity in the spacetime M2, thus the novel shadowradius observed in the spacetime M1 is rsh1 = bR1 .(B2). Next, we pay our attention to the case k > 1.

Similarly, we find bR2 > bc2 for R > max2k, 3. Thus, forR > 3k, the outgoing geodesics will go to infinity in thespacetime M2, that is, the shadow radius is rsh1 = bR1 .While, for max2k, 3 < R < 3k, the critical impact

parameter bc2 = 3√

3k corresponds to Bc1 = 3√

3k√

R−2kR−2

which is same with the case 2 < R ≤ 3 in surprise. Andafter some simple analysis, we find Bc1 < bR1 as expected.Therefore, in this case, the asymmetry of the spactimeM results in a novel shadow very interestingly, which isalso worth in-depth study.

Discussion.—To summarize, by analyzing the behaviorof null geodesics passing through the throat of the staticthin-shell wormhole model, we identify novel shadows re-sulting from the asymmetry between the two sides of thethin shell and their sizes are not dependent on the pho-ton sphere in the observer’s spacetime. When the ob-server’s spacetime contains a phono sphere, if a staticobserver lives on the side with a smaller mass parame-ter for a wormhole spacetime, taking the backward ray-tracing point of view, all the ingoing geodesics with im-pact parameters smaller than the critical one 3

√3 will

pass through the thin shell and become outgoing in theother side of the spacetime. However, different from the“one way” property of black holes, some of them abovea new critical impact parameter will stop at the turningpoint, bounce back to the original spacetime and comeinto the eyes of the static observer due to the asymme-try. Thus, the novel shadow is always smaller than theanalogue of black holes. The difference can be very signif-icant to overcome the uncertainty in the measurementsof black hole mass and its distance from observers sothat observing the thin-shell wormholes directly throughEvent horizon Telescope becomes feasible. While, theobserver lives on the side with a larger mass parameterfor a wormhole spacetime, the observational shadow is assame as that of a black hole, so one cannot discern themby imaging shadow.

Moreover, when the radius of the thin shell is largerthan that of the photon sphere, we also show that thereare novel shadows even though the observer’s spacetime

5

does not contain the photon sphere, which is an impor-tant complement to the related existing knowledge.

The mechanism for predicting this novel shadow of thethin-shell wormhole is very simple and elegant, we be-lieve it could be a universal property. Thus it would beextremely interesting to see if this novel shadow can befound in other wormhole models in general relativity ormodified theories of gravity. Furthermore, searching forthis novel shadow by Event Horizon Telescope is also anexciting task in the future to know whether the thin-shell wormhole exists, or not. Concerning the EHT2017observations [1], as analyzed in [19] the compact objectswhose shadows exhibit qualitatively deviation from those

of black holes can be ruled out for M87* (however, see[20] for exception). So the thin-shell wormhole modelscan be strongly constrained with the present observa-tions. However it may still has the chance to discoverwormhole by EHT according to the novel shadow featurein the future.

The work is in part supported by NSFC Grant No.11335012, No. 11325522 and No. 11735001. MG andPCL are also supported by NSFC Grant No. 11947210.And MG is also funded by China Postdoctoral ScienceFoundation Grant No. 2019M660278 and 2020T130020.PCL is also funded by China Postdoctoral Science Foun-dation Grant No. 2020M670010. CYZ is supported byNSFC Grant No. 11947067.

[1] K. Akiyama et al. [Event Horizon Telescope Collabora-tion], Astrophys. J. Lett. 875, L1 (2019).

[2] J. L. Synge, Mon. Not. Roy. Astron. Soc. 131, no. 3, 463(1966).

[3] J. M. Bardeen, “Timelike and null geodesies in the Kerrmetric,” in Proceedings, Ecole d’Ete de Physique Theo-rique: Les Astres Occlus, eds. C. Witt and B. Witt (LesHouches, France, 1973), p. 215-239.

[4] M. Visser, “Lorentzian wormholes: From Einstein toHawking,” Woodbury, USA: AIP (1995) 412 p.

[5] O. Sarbach and T. Zannias, AIP Conf. Proc. 1473, no.1, 223 (2012).

[6] C. Bambi, Phys. Rev. D 87, 107501 (2013).[7] P. G. Nedkova, V. K. Tinchev and S. S. Yazad-

jiev, Phys. Rev. D 88, no.12, 124019 (2013)doi:10.1103/PhysRevD.88.124019 [arXiv:1307.7647 [gr-qc]].

[8] T. Ohgami and N. Sakai, Phys. Rev. D 91,no.12, 124020 (2015) doi:10.1103/PhysRevD.91.124020[arXiv:1704.07065 [gr-qc]].

[9] F. Lamy, E. Gourgoulhon, T. Paumard and F. H. Vin-cent, Class. Quant. Grav. 35, no.11, 115009 (2018)doi:10.1088/1361-6382/aabd97 [arXiv:1802.01635 [gr-

qc]].[10] F. H. Vincent, M. Wielgus, M. A. Abramowicz, E. Gour-

goulhon, J. P. Lasota, T. Paumard and G. Perrin,[arXiv:2002.09226 [gr-qc]].

[11] H. G. Ellis, J. Math. Phys. 14, 104 (1973).[12] M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395

(1988).[13] M. Visser, Phys. Rev. D 39, 3182 (1989).[14] M. Visser, Nucl. Phys. B 328, 203 (1989).[15] N. M. Garcia, F. S. N. Lobo and M. Visser, Phys. Rev.

D 86, 044026 (2012) doi:10.1103/PhysRevD.86.044026[arXiv:1112.2057 [gr-qc]].

[16] D. C. Dai and D. Stojkovic, “Observing a Wormhole,”Phys. Rev. D 100, no.8, 083513 (2019).

[17] D. Langlois, K. i. Maeda and D. Wands, Phys. Rev. Lett.88, 181301 (2002).

[18] K. i. Nakao, T. Uno and S. Kinoshita, Phys. Rev.D 88, 044036 (2013) doi:10.1103/PhysRevD.88.044036[arXiv:1306.6917 [gr-qc]].

[19] K. Akiyama et al. [Event Horizon Telescope Collabora-tion], Astrophys. J. 875, no. 1, L5 (2019).

[20] C. Bambi, K. Freese, S. Vagnozzi and L. Visinelli, Phys.Rev. D 100, no.4, 044057 (2019).